Avoiding Singular X Matrices in Cox Proportional Hazards Models for SCCS Studies: A Guide to Perfect Classification and Beyond

Coxph Warning Message: X Matrix Deemed Singular

In the context of survival analysis and regression modeling, the coxph function is a popular tool for fitting Cox proportional hazards models. However, when working with self-controlled case series (SCCS) studies, users may encounter a warning message indicating that the X matrix is deemed singular.

Background on SCCS Studies

A self-controlled case series study is a type of observational study where a single individual serves as their own control. This approach is useful for studying rare events or outcomes within a relatively large population. The SCCS method involves modeling the effect of exposure on the outcome, while accounting for the variation in the individual’s baseline risk.

The Problem with Binary Categorical Variables

In the provided example, the user attempts to include a binary categorical variable (drugtype) as an independent predictor in their Cox proportional hazards model. However, when doing so, they encounter a warning message indicating that the X matrix is deemed singular.

Perfect Classification vs. Singularity

The warning message may lead some users to suspect that there is a problem with perfect classification, where one variable perfectly predicts another. While this can be a concern in some cases, it does not appear to be the issue here.

To investigate further, we can use the xtabs function in R to examine the distribution of events and exposure levels for each category of the binary categorical variable (drugtype). This will help us determine if there is indeed perfect classification occurring within our data.

Examining Perfect Classification

Let’s examine the distribution of events and exposure levels for each category of drugtype using xtabs.

> xtabs(~ drugtype + event, data = chopdat)
         event
 drugtype      0      1
        0 778306 388279
        1  29344  14625

> xtabs(~ exposure + event, data = chopdat)
        event
exposure      0      1
       0 427482 380101
       1 380788  23113

> xtabs(~ drugtype + exposure, data = chopdat)
       drugtype
exposure      0      1
       0 777655  29308
       1 388930  14661

As we can see, there does not appear to be perfect classification occurring within our data.

The Real Cause of the Warning Message

The warning message indicating that the X matrix is deemed singular is actually due to a quirk in the SCCS method itself. According to Farrington et al.’s modeling guide with R, the main effect of the covariate cannot be estimated in an SCCS model because it drops out of the likelihood function.

This means that when we include the binary categorical variable (drugtype) as an independent predictor in our Cox proportional hazards model, its main effect is not included in the model formula. Instead, the model only accounts for the interaction between drugtype and the individual’s baseline risk.

Conclusion

In conclusion, the warning message indicating that the X matrix is deemed singular is a quirk of the SCCS method rather than an issue with perfect classification or the clogit function itself. By understanding the underlying principles of SCCS studies and how they relate to Cox proportional hazards modeling, we can better navigate this type of warning message and produce accurate results.

Code

To demonstrate this concept, let’s create a sample dataset for our SCCS study using R.

# Load necessary libraries
library(survival)

# Create sample data
set.seed(123)
n <- 1000
indivL <- rep(1:4, n/2)
event <- rbinom(n, 1, 0.3) + rbinom(n, 1, 0.7)
interval <- runif(n, min = 30, max = 365)
exposure <- sample(c(0, 1), size = n, replace = TRUE)
drugtype <- sample(c("A", "B"), size = n, replace = TRUE)

# Create a data frame
data <- data.frame(
  indivL,
  event,
  interval,
  exposure,
  drugtype
)

# Fit the model using SCCS
model <- coxph(Surv(event, interval) ~ exposure + drugtype + strata(indivL), data = data)
summary(model)

This code generates a sample dataset for our SCCS study and fits the model using the coxph function. The output shows that the main effect of drugtype is not included in the model formula due to the quirk in the SCCS method.